NLC-2 graph recognition and isomorphism

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NLC-2 graph recognition and isomorphism

Vincent Limouzy, Fabien de Montgolfier, Michaël Rao

To cite this version:

Vincent Limouzy, Fabien de Montgolfier, Michaël Rao. NLC-2 graph recognition and isomorphism.

Graph-Theoretic Concepts in Computer Science 33rd International Workshop, WG 2007, Dornburg,

Germany, June 21-23, 2007., Jun 2007, Dornburg, Germany. pp.86-98, �10.1007/978-3-540-74839-7_9�.

�hal-00134605�

hal-00134605, version 1 - 2 Mar 2007

Vin ent Limouzy

1

Fabiende Montgoler

1

Mi haël Rao

1

Abstra t

NLC-widthisavariantof lique-widthwithmanyappli ationingraphalgorithmi . This paperisdevotedtographsofNLC-widthtwo. Aftergivingnewstru tural propertiesof the lass,weproposea

O(n

2

_m)

-timealgorithm,improvingJohansson'salgorithm[14℄.Moreover, ouralogrithmissimpletounderstand. Theabovepropertiesandalgorithmallowustopropose arobust

O(n

2

_m)

-timeisomorphismalgorithmforNLC-2graphs.Asfarasweknow,itisthe rstpolynomial-timealgorithm.

1 Introdu tion

NLC-width is a graph parameter introdu ed by Wanke [16 ℄. This notion is tightly related to

lique-width introdu ed by Cour elle et al. [2℄. Both parameters were introdu ed to generalise the well known tree-width. The motivation on resear h about su h width parameter is that, whenthewidth(NLC-, lique-ortree-width) isbounded bya onstant,thenmanyNP- omplete

problems anbesolvedinpolynomial(even linear)time, ifthede omposition isprovided. Su h parameters give insights on graph stru tural properties. Unfortunately, nding the minimumNLC-width of the graphwasshownto be NP-hard byGurskiet al. [12 ℄. Some results

however are known. Let NLC-

k

be the lass of graph of NLC width bounded by

k

. NLC-1 is exa tly the lass of ographs. Probe- ographs, bi- ographs and weak-bisplit graphs [9℄ belong to NLC-2. Johansson[14 ℄ proved that re ognising NLC-2graphs is polynomial and provided an

O(n

4

_log(n))

re ognitionalgorithm. Complexityforre ognitionofNLC-

k

,

k

≥ 3

,isstillunknown. In thispaperwe improve Johansson's resultdownto

O(n

2

_m)

. Our approa h relies ongraph de ompositions. We establish the tight links thatexist between NLC-2graphs andtheso- alled modularde omposition, splitde omposition, and bi-joinde omposition.

NLC-2 an be dened as a graph olouring problem. Unlike NLC-

k

lasses, for

k

≥ 3

, re olouringisuselessforprimeNLC-2graphs. Thatallowustoproposea anoni alde omposition

ofbi- oloured NLC-2graphs, denedas ertainbi- oloured splitoperations. Thisde omposition an be omputed in

O(nm)

time if the olouring is provided. If a graph is prime, there using splitandbi-joinde ompositions,weshowthatthereisatmost

O(n)

olouringsto he k. Finally, modular de omposition properties allowto redu eNLC-2 graph de omposition to primeNLC-2 graphde omposition. Se tion 3 explains this

O(n

2

_m)

-time de omposition algorithm.

In Se tion 4is proposed an isomorphism algorithm. Using modular, split andbi-join de om-positions and the anoni alNLC-2de omposition, isomorphism between two NLC-2graphs an

be tested in

O(n

2

_m)

time.

2 Preliminaries

Agraph

G

= (V, E)

ispairofasetofverti es

V

andasetofedges

E

. Foragraph

G

,

V

(G)

denote itssetofverti es,

E(G)

its setofedges,

n(G) = |V (G)|

and

m(G) = |E(G)|

(or

V

,

E

,

n

and

m

if

1

LIAFA,Université Paris7. {limouzy,fm,rao}liafa.jussieu. fr. Resear hsupportedbythe Fren hANR proje tGraphDe ompositionsandAlgorithms(GRAAL)

thegraphis lear inthe ontext).

N

(x) = {y ∈ V : {x, y} ∈ E}

denotestheneighbourhood ofthe vertex

x

,and

N[x] = N (v) ∪ {v}

. For

W

⊆ V

,

G[W ] = (W, E ∩ W

2

₎

denotethegraph indu ed by

W

. Let

A

and

B

betwodisjointsubsetsof

V

. Thenwenote

A

1

B

ifforall

(a, b) ∈ A × B

,then

{a, b} ∈ E

,andwenote

A

B

0 ifforall

(a, b) ∈ A × B

,then

{a, b} 6∈ E

. Two graphs

G

= (V, E)

and

G

′

_{= (V}

′

_{, E}

′

₎

are isomorphi (noted

G

≃ G

′

) ifthere is a bije tion

ϕ

: V → V

′

su h that

{x, y} ∈ E ⇔ {ϕ(x), ϕ(y)} ∈ E

′

,for all

u, v

∈ V

.

A

k

-labelling (or labelling) is a fun tion

l

: V → {1, . . . , k}

. A

k

-labelled graph is a pair of a graph

G

= (V, E)

and a

k

-labelling

l

on

V

. It is denoted by

(G, l)

or by

(V, E, l)

. Two labelled graphs

(V, E, l)

and

(V

′

_{, E}

′

_{, l}

′

₎

are isomorphi if there is a bije tion

ϕ

: V → V

′

su h that

{u, v} ∈ E ⇔ {ϕ(x), ϕ(y)} ∈ E

′

and

l(u) = l

′

_(ϕ(u))

for all

u, v

∈ V

.

NLC-

k

lasses. Let

k

be apositive integer. The lassof NLC-

k

graphsis dened re ursively bythe following operations.

•

For all

i

∈ {1, . . . , k}

,

·(i)

is inNLC-

k

,where

·(i)

is thegraph withone vertexlabelled

i

.

•

Let

G

1

= (V

1

, E

1

, l

1

)

and

G

2

= (V

2

, E

2

, l

2

)

be NLC-

k

and let

S

⊆ {1, . . . , k}

2

. Then

G

₁

×

S

G

2

isin NLC-

k

,where

G

1

×

S

G

2

= (V, E, l)

with

V

= V

1

∪ V

2

,

E

= E

1

∪ E

2

∪ {{u, v} : u ∈ V

1

, v

∈ V

2

,

(l

1

(u), l

2

(v)) ∈ S}

andfor all

u

∈ V

,

l(u) =

(

l

₁

(u)

if

u

∈ V

1

l

2

(u)

if

u

∈ V

2

.

•

Let

R

: {1, . . . , k} → {1, . . . , k}

and

G

= (V, E, l)

be NLC-

k

. Then

ρ

R

(G)

is in NLC-

k

, where

ρ

R

(G) = (V, E, l

′

₎

su hthat

l

′

_{(u) = R(l(u))}

for all

u

∈ V

.

A graphis NLC-

k

ifthere isa

k

-labelling of

G

su h that

(G, l)

is inNLC-

k

. A

k

-labelled graph isNLC-

k ρ

-free ifit an be onstru ted withoutthe

ρ

R

operation.

Modules and modularde omposition. Amodule inagraph isa non-emptysubset

X

⊆ V

su h that for all

u

∈ V \ X

, then either

N

(u) ∩ X = ∅

or

X

⊆ N (u)

. A module is trivial if

|X| ∈ {1, |V |}

. A graph is prime (w.r.t. modular de omposition) if all its modules are trivial. Two sets

X

and

X

′

overlap if

X

∩ X

′

,

X

\ X

′

and

X

′

_{\ X}

arenon-empty. A module

X

isstrong if there is no module

X

′

su h that

X

and

X

′

overlap. Let

M

′

_(G)

be the set of modules, let

M(G)

be theset of strong modules of

G

, and let

P(G) = {M

1

, . . . , M

k

}

be themaximal (w.r.t. in lusion)members of

M(G) \ {V }

.

Theorem 1. [11℄Let

G

= (V, E)

be a graph su h that

|V | ≥ 2

. Then:

•

if

G

is not onne ted, then

P(G)

isthe set of onne ted omponents of

G

,

•

if

G

is not onne ted, then

P(G)

isthe set of onne ted omponents of

G

,

•

if

G

and

G

are onne ted, then

P(G)

is a partition of

V

and is formed with the maximal members of

M

′

_{\ {V }}

.

Inall ases,

P(G)

isa partitionof

V

,and

G

anbede omposedinto

G[M

1

], . . . , G[M

k

]

. The hara teristi graph

G

∗

of a graph

G

is the graph of vertex set

P(G)

and two

P, P

′

_{∈ P(G)}

are adja ent if there is an edge between

P

and

P

′

in

G

(and so there is no non-edges sin e

P

and

P

′

are two modules). The re ursive de omposition of a graph by this operation gives

de omposition tree are exa tly the strong modules, so in the following we make no distin tion between the modular de omposition of

G

and

M(G)

. Note that

|M(G)| ≤ 2 × n − 1

. For

M

∈ M(G)

, let

G

M

= G[M ]

and

G

∗

M

its hara teristi graph.

Lemma 2. [14℄ Let

G

be a graph.

G

is NLC-

k

if and only if every hara teristi graph in the modular de omposition of

G

isNLC-

k

.

Moreover,aNLC-

k

expressionfor

G

anbeeasily onstru tedfromthemodularde omposition andfrom NLC-

k

expressions ofprimegraphs. Onprimegraphs, NLC-2re ognition is easier: Lemma 3. [14℄ Let

G

be a primegraph. Then

G

isNLC-2 if andonly if there is a

2

-labelling

l

su h that

(G, l)

isNLC-2

ρ

-free.

Bi-partitive family. A bipartition of

V

is a pair

{X, Y }

su h that

X

∩ Y = ∅

,

X

∪ Y = V

and

X

and

Y

are both non empty. Two bipartitions

{X, Y }

and

{X

′

_{, Y}

′

_}

overlap if

X

∩ Y

,

X

∩ Y

′

,

X

′

_{∩ Y}

and

X

′

_{∩ Y}

′

are nonempty. A family

F

ofbipartitions of

V

is bipartitive if(1) forall

v

∈ V

,

{{v}, V \ {v}} ∈ F

and(2)for all

{X, Y }

and

{X

′

_{, Y}

′

_}

in

F

su hthat

{X, Y }

and

{X

′

_{, Y}

′

_}

overlap,then

{X ∩ X

′

_{, Y}

_{∪ Y}

′

_}

,

{X ∩ Y

′

_{, Y}

_{∪ X}

′

_}

,

{Y ∩ X

′

_{, X}

_{∪ Y}

′

_}

,

{Y ∩ Y

′

_{, X}

_{∪ X}

′

_}

and

{X∆X

′

_{, X∆Y}

′

_}

arein

F

(where

X∆Y = (X \ Y ) ∪ (Y \ X)

). Bipartitive families arevery lose to partitive families[1 ℄,whi h generalisepropertiesof modules inagraph.

Amember

{X, Y }

ofa bipartitive family

F

isstrong ifthereisno

{X

′

_{, Y}

′

_}

su h that

{X, Y }

and

{X

′

_{, Y}

′

_}

overlap. Let

T

beatree. For anedge

e

inthetree,

{C

1

e

, C

e

2

}

denotethebipartition ofleavesof

T

su hthattwoleavesareinthesamesetifandonlyifthepathbetween themavoids

e

. Similarly,foran internalnode

α

,

{C

1

α

, . . . , C

d(α)

α

}

denotethepartition ofleavesof

T

su hthat twoleavesareinthe same setifand only ifthepath between them avoid

α

.

Theorem 4. [3℄ Let

F

be a bipartitive family on

V

. Then there is an unique unrooted tree

T

, alled the representative treeof

F

, su h that the set of leaves of

T

is

V

, the internal nodes of

T

are labelled degenerateor prime, and

- for every edge

e

of

T

,

{C

1

e

, C

e

2

}

is a strong member of

F

, and there is no other strong member in

F

,

- for every node

α

labelled degenerate,and for every

∅ ( I ( {1, . . . , d(α)}

,

{∪

i∈I

C

α

i

, V

\ ∪

i∈I

C

α

i

}

isin

F

,and there is no other member in

F

.

Split de omposition. A split in a graph

G

= (V, E)

is a bipartition

{X, Y }

of

V

su h that thesetof verti es in

X

having a neighbour in

Y

have thesame neighbourhood in

Y

(i.e., for all

u, v

∈ X

su hthat

N

(u) ∩ Y 6= ∅

and

N

(v) ∩ Y 6= ∅

,then

N

(u) ∩ Y = N (v) ∩ Y

). A o-split ina graph

G

is a splitin

G

. Thefamily of splitina onne ted graph is abipartitive family [4℄. The splitde ompositiontree isthe representativetreeof thefamily ofsplits,and anbe omputed in linear time[5 ℄. Let

α

be an internal node of the split de omposition tree of a onne ted graph

G

. For all

i

∈ {1, . . . , d(α)}

let

v

i

∈ C

i

α

su h that

N

(v

i

) \ C

i

α

6= ∅

. Sin e

G

is onne ted, su h a

v

i

alwaysexists.

G[{v

1

, . . . , v

d(α)

}]

denotethe hara teristi graph of

α

. The hara teristi graph ofa degeneratenodeis a omplete graph or astar [4 ℄.

Bi-joinde omposition. Abi-join inagraphisabipartition

{X, Y }

su hthatforall

u, v

∈ X

,

{N (u)∩ Y, Y \N (u)} = {N (v)∩ Y, Y \N (v)}

. Thefamilyofbi-joinsinagraphisbipartitive. The bi-joinde omposition tree istherepresentativetreeofthefamilyofbi-joins,and anbe omputed inlinear time [7, 8℄. Let

α

be an internal node of the bi-join de omposition tree of a graph

G

. For all

i

∈ {1, . . . , d(α)}

let

v

i

∈ C

i

α

.

G[{v

1

, . . . , v

d(α)

}]

denotethe hara teristi graph of

α

. The hara teristi graph of a degenerate node is a omplete bipartite graph or a disjoint union of two omplete graphs [7,8℄.

3 Re ognition of NLC-2 graphs

3.1 NLC-2

ρ

-free anoni al de omposition

Inthis se tion,

G

= (V, E, l)

isa 2-labelled graph su h thatevery mono- olouredmodule (i.e. a module

M

su h that

∀v, v

′

_{∈ M}

,

l(v) = l(v

′

₎

)hassize

1

. A ouple

(X, Y )

isa ut if

X

∪ Y = V

,

X

∩ Y = ∅

,

X

6= ∅

and

Y

6= ∅

. Let

S

⊆ {1, 2} × {1, 2}

. A ut

(X, Y )

is a

S

- ut of

G

iffor all

u

∈ X

and

v

∈ Y

, then

{u, v} ∈ E

if and only if

(l(u), l(v)) ∈ S

. For

S

⊆ {1, 2} × {1, 2}

let

F

S

(G)

be thesetof

S

- utof

G

.

Denition 5 (Symmetry). We say that

S

∈ {1, 2} × {1, 2}

is symmetri if

(1, 2) ∈ S ⇐⇒

(2, 1) ∈ S

,otherwisewesaythat

S

isnon-symmetri .

Denition 6 (Degenerate property). Afamily

F

of utshasthedegenerate property ifthere isapartition

P

of

V

su hthatfor all

∅ ( X ( P

,

(

S

X

∈X

X,

S

Y

∈P\X

Y

)

isin

F

,andthere isno others utin

F

.

Lemma 7. For every symmetri

S

⊆ {1, 2} × {1, 2}

,

F

S

(G)

has the degenerate property. Proof. The family

F

{}

(G)

has the degenerate property sin e

(X, Y )

is a

{}

- ut if and only if there is no edges between

X

and

Y

(

P

is exa tly the onne ted omponents). For

W

⊆ V

, let

G|W = (V, E∆W

2

_{, l)}

. For

i

∈ {1, 2}

let

V

i

= {v ∈ V : l(v) = i}

. Let

G

1

= G|V

1

,

G

2

= G|V

2

and

G

₁₂

= (G|V

1

)|V

2

.

• F

{(1,1)}

(G) = F

{}

(G

1

)

,

F

{(2,2)}

(G) = F

{}

(G

2

)

,

F

{(1,1),(2,2)}

(G) = F

{}

(G

12

)

,

• F

{(1,1),(1,2),(2,1),(2,2)}

(G) = F

{}

(G)

,

F

{(1,2),(2,1),(2,2)}

(G) = F

{}

(G

1

)

,

F

{(1,1),(1,2),(2,1)}

(G) = F

{}

(G

2

)

,

F

{(1,2),(2,1)}

(G) = F

{}

(G

12

)

.

Thus for every symmetri

S

⊆ {1, 2} × {1, 2}

,

F

S

(G)

hasthedegenerateproperty.

Denition 8 (Linear property). A family

F

of uts has thelinear property iffor all

(X, Y )

and

(X

′

_{, Y}

′

₎

in

F

,either

X

⊆ X

′

or

X

′

_{⊆ X}

.

Lemma 9. For every non-symmetri

S

⊆ {1, 2} × {1, 2}

,

F

S

(G)

has the linear property. Proof. Case

S

= {(1, 2)}

: suppose that

X

\ X

′

and

X

′

_{\ X}

are both non-empty. Then if

u

∈

X

\ X

′

is labelled

1

and

v

∈ X

′

_{\ X}

is labelled

2

,

u

and

v

hasto be adja ent and non-adja ent, ontradi tion. Thus

X

\ X

′

and

X

′

_{\ X}

aremono- oloured. Nowsupposew.l.o.g. thatallverti es in

X∆X

′

are labelled

1

. Then

X∆X

′

is adja ent to all verti es labelled

2

in

Y

∩ Y

′

and non adja ent to all verti es labelled

1

in

Y

∩ Y

′

. Moreover

X∆X

′

is non adja ent to all verti es in

X

∩ X

′

. Thus

X∆X

′

is a mono- oloured module, and

|X∆X

′

_{| ≥ 2}

. Contradi tion. For others non-symmetri

S

,webring ba kto ase

{(1, 2)}

like intheproof oflemma 7.

Input: A

2

-labelled graph

G

= (V, E, l)

Output: ANLC-2

ρ

-freede omposition tree,or failif

G

is not NLC-2

ρ

-free if

|V | = 1

thenreturnthe leaf

·(l(v))

(where

V

= {v}

)

1

Let

S

be the setof subsetsof

{1, 2} × {1, 2}

and

σ

bethelexi ographi orderof

S

2

forea h

S

∈ S

w.r.t.

σ

do 3

Compute

P

S

(G)

,and

P

′

S

(G)

if

S

is non-symmetri (see algorithm 2) 4

if

|P

S

(G)| > 1

then 5

Create anew

×

S

node

β

6

forea h

P

∈ P

S

(G)

(w.r.t.

P

′

S

(G)

if

S

is non-symmetri ) do 7

make NLC-2

ρ

-free de omposition tree of

G[P ]

bea hildof

β

. 8

return the tree rooted at

β

9

fail withNot NLC-2

ρ

-free 10

Algorithm 1: Computationof the NLC-2

ρ

-free anoni alde omposition tree

For

S

⊆ {1, 2} × {1, 2}

, let

P

S

(G)

denote the unique partition of

V

su h that (1) for all

(X, Y ) ∈ F

S

(G)

and

P

∈ P

S

(G)

,

P

⊆ X

or

P

⊆ Y

,and (2) for all

P, P

′

_{∈ P}

,

P

6= P

′

,there is a

(X, Y ) ∈ F

S

(G)

su h that

P

⊆ X

and

P

′

_{⊆ Y}

,or

P

⊆ Y

and

P

′

_{⊆ X}

. For a non-symmetri

S

∈ {1, 2} × {1, 2}

, let

P

′

S

(G) = (P

1

, . . . , P

k

)

denote the unique orderingof elements in

P

S

(G)

su hthat for all

(X, Y ) ∈ F

S

(G)

,thereis a

l

su hthat

X

= ∪

i∈{1,...,l}

P

i

.

Lemma 10. If

G

is in NLC-2

ρ

-free, then there is a

S

⊆ {1, 2} × {1, 2}

su h that

F

S

(G)

is non-empty.

Proof. If

G

is NLC-2

ρ

-free, thenthereis a

S

⊆ {1, 2} × {1, 2}

,and two graphs

G

1

and

G

2

su h that

G

= G

1

×

S

G

2

. Thus

(V (G

1

), V (G

2

)) ∈ F

S

(G)

and

F

S

(G)

isnon empty.

Lemma 11. Let

G

= (V, E, l)

2-labelled graph andlet

S

⊆ {1, 2} × {1, 2}

. If

G

is NLC-2

ρ

-free andhasnomono- olourednon-trivialmodule,thenforall

P

∈ P

S

(G)

,

G[P ]

hasnomono- oloured non-trivialmodule.

Proof. If

M

is amono- oloured moduleof

G[P ]

,then

M

isa mono- olouredmodule of

G

. Con-tradi tion.

Lemma 12. Let

G

= (V, E, l)

2-labelled graph and let

S

⊆ {1, 2} × {1, 2}

. Then

G

is NLC-2

ρ

-free if andonlyif for all

P

∈ P

S

(G)

,

G[P ]

isNLC-2

ρ

-free.

Proof. Theonlyif isimmediate. Nowsupposethatfor all

P

∈ P

S

(G)

,

G[P ]

isNLC-2

ρ

-free. If

S

issymmetri ,let

P

S

(G) = {P

1

, . . . , P

|P

S

(G)|

}

. Then

G

= ((G[P

1

]×

S

G[P

2

])×

S

. . .

×

S

G[P

|P

S

(G)|

]

, and

G

is NLC-2

ρ

-free. Otherwise, if

S

isnon-symmetri , let

P

′

S

(G) = (P

1

, . . . , P

|P

S

(G)|

)

. Then

G

= ((G[P

1

] ×

S

G[P

2

]) ×

S

. . .

×

S

G[P

|P

S

(G)|

]

,and

G

isNLC-2

ρ

-free.

The NLC-2

ρ

-free de omposition tree of a

2

-labelled graph

G

is a rooted tree su h that the leavesare theverti es of

G

,and the internal nodesarelabelled by

×

S

,with

S

⊆ {1, 2} × {1, 2}

. An internal node is degenerated if

S

is symmetri , and linear if

S

is non-symmetri . By lemmas10,11 and12,

G

isNLC-2

ρ

-freeifand onlyifithasaNLC-2

ρ

-free de ompositiontree. Thisde ompositiontreeisnot unique. Butwe andene a anoni al de omposition tree ifwex

atotalorderonthesubsetsof

{1, 2} × {1, 2}

(forexample,thelexi ographi order). Iftwographs are isomorphi , then they have the same anoni al de omposition tree. Algorithm 1 omputes the anoni alde ompositiontree ofa

2

-labelled prime graph, orfailsif

G

isnot NLC-2

ρ

-free.

Algorithm 2 omputes

P

S

and

P

′

S

for a

2

-labelled primegraph

G

and

S

⊆ {1, 2} × {1, 2}

in lineartime. Weneedsomeadditionaldenitionsforthisalgorithmanditsproofof orre tness. A

Input: A

2

-labelled graph

G

,and

S

⊆ {1, 2} × {1, 2}

Output:

P

S

if

S

issymmetri ,

P

′

S

if

S

is non-symmetri

V

i

← {v : v ∈ V

and

l(v) = i}

; 1

if

(1, 1) ∈ S

then

C

1

←

o- onne ted omponentsof

G[V

1

]

; 2

else

C

1

←

onne ted omponents of

G[V

1

]

; 3

if

(2, 2) ∈ S

then

C

2

←

o- onne ted omponentsof

G[V

2

]

; 4

else

C

2

←

onne ted omponents of

G[V

2

]

; 5

B = (C

1

,

C

2

, E

j

, E

m

) ←

thebipartite trigraphbetween theelements of

C

1

and

C

2

; 6

if

S

∩ {(1, 2), (2, 1)} = ∅

then 7

return onne ted omponents of

(C

1

,

C

2

, E

j

∪ E

m

)

8

else if

S

∩ {(1, 2), (2, 1)} = {(1, 2), (2, 1)}

then 9

return onne ted omponents of the bi- omplement of

(C

1

,

C

2

, E

j

)

10

else Sear h allsemi-joins of

B

(see appendix) ; 11

Algorithm 2: Computationof

P

S

and

P

′

S

bipartite graphisatriplet

(X, Y, E)

su hthat

E

⊆ X ×Y

. Thebi- omplementofabipartitegraph

(X, Y, E)

isthebipartitegraph

(X, Y, (X × Y ) \ E)

. Abipartitetrigraph(BT)isabipartitegraph withtwotypesofedges: thejoin edgesandthemixed edges. Itisdenotedby

B = (X, Y, E

j

, E

m

)

where

E

j

are the set of join edges, and

E

m

the set of mixed edges. A BT-module in a BT is a

M

⊆ X

or

M

⊆ Y

su h that

M

is a module in

(X, Y, E

j

)

and there is no mixed edges between

M

and

(X ∪ Y ) \ M

. For

v

∈ X ∪ Y

, let

N

j

(v) = {u ∈ X ∪ Y : {u, v} ∈ E

j

}

and

N

m

(v) = {u ∈ X ∪ Y : {u, v} ∈ E

m

}

. Let

d

j

(v) = |N

j

(v)|

and

d

m

(v) = |N

m

(v)|

. A semi-join inaBT

(X, Y, E

j

, E

m

)

isa ut

(A, B)

of

X

∪ Y

,su hthatthere isno edgesbetween

A

∩ Y

and

B

∩ X

,and thereisonly join edges between

A

∩ X

and

B

∩ Y

.

Inalgorithm2,

B

isobtainedfromthegraph

G

. Verti esof

X

orrespondtosubsetsofverti es labelled

1

in

G

, and verti es of

Y

orrespond to subsets of verti es labelled

2

. There is a join edgebetween

M

and

M

′

in

B

if

M

1

M

′

in

G

,and there isa mixed edgebetween

M

∈ X

and

M

′

∈ Y

in

B

ifthereis at leastan edgeand anon-edge between

M

and

M

′

in

G

. Su h agraph

B

an easily be built in linear time from a given graph

G

. It su es to onsider a list and an array bounded by the number of omponent in

G

with the same olour. The following lemmas are lose toobservations in[9℄,but dealwithBT insteadofbipartite graphs(proofsaregiven in appendix).

Lemma 13. Let

G

= (X, Y, E

j

, E

m

)

be a BT su h that every BT-module has size

1

. Let

(x

1

, . . . , x

|X|

)

be

X

sorted by

(d

j

(x), d

m

(x))

in lexi ographi de reasing order. If

(A, B)

is a semi-joinof

G

, then there is a

k

∈ {0, . . . , |X|}

su h that

A

∩ X = {x

1

, . . . , x

k

}

.

Lemma 14. Let

k

∈ {0, . . . , |X|}

and

k

′

_{∈ {0, . . . , |Y |}}

. Then

(A, (X ∪ Y ) \ A)

, where

A

=

{x

1

, . . . , x

k

, y

1

, . . . , y

k

′

}

,isasemi-joinof

G

ifandonlyif

P

k

i=1

d

j

(x

i

)−

P

k

′

i=1

d

j

(y

i

) = k×(|Y |−k

′

)

and

P

k

i=1

d

m

(x

i

) −

P

k

′

i=1

d

m

(y

i

) = 0

.

Theorem 15. Algorithm 2 is orre t and runs in linear time.

Proof. Corre tness: Supposethat

(A, B)

isa

S

- ut. If

(1, 1) 6∈ S

,thenthereisnoedgebetween

A∩V

1

and

B

∩V

1

,thus

(A, B)

annot uta omponent

C

1

(andsimilarlyfor

(1, 1) ∈ S

,andfor

C

2

). Now we work on the BT

B = (C

1

,

C

2

, E

j

, E

m

)

. If

S

∩ {(1, 2), (2, 1)} = ∅

,then

S

- uts orrespond exa tly to onne ted omponents of

B

, and if

S

∩ {(1, 2), (2, 1)} = {(1, 2), (2, 1)}

then

S

- uts orrespond exa tly to onne ted omponents of the BT of

G

,whi h is

(C

1

,

C

2

,

(C

1

× C

2

) \ (E

j

∪

E

m

), E

m

)

. Finally,if

S

is non-symmetri ,

S

- uts orrespond to semi-joinsof

B

(seeappendix).

inlineartime [13 , 6 ℄. The instru tions onlines[2-5,8℄ an bedone witha BFS ona graph or its omplement. Itis easy to see thatwe an do a BFS on thebi- omplement inlineartime (like a BFS on a omplement graph, withtwo vertex lists for

X

and

Y

), so instru tion line 10 an be doneinlineartime. Finally,the operationsat line 11aredone inlineartime(seeappendix).

Theseresults an be summarizedas:

Theorem16. Algorithm1 omputesthe anoni alNLC-2

ρ

-freede ompositiontree ofa2-labelled graph in

O(nm)

time.

3.2 NLC-2 de omposition of a prime graph

Inthis se tion,

G

isan unlabelled prime (w.r.t. modularde omposition) graph, with

|V | ≥ 3

. Denition17(

2

-bimodule). Abipartition

{X, Y }

of

V

isa

2

-bimodule if

X

anbepartitioned into

X

1

and

X

2

,and

Y

into

Y

1

and

Y

2

su hthatforall

(i, j) ∈ {1, 2}× {1, 2}

,theneither

X

i

0

Y

j

or

X

i

1

Y

j

. It is easy to see that if

{X, Y }

is a

2

-bimodule ifand only if

{X, Y }

is a split, a o-split or abi-join. Moreover, if

min(|X|, |Y |) > 1

then

{X, Y }

annot be both of them in the same time(sin e

G

isprime).

Let

l

: V → {1, 2}

be a 2-labelling. Then

s(l)

denote the 2-labelling on

V

su h that for all

v

∈ V

,

s(l)(v) = 1

ifandonly if

l(v) = 2

.

Denition 18 (Labelling indu ed by a

2

-bimodule). Let

{X, Y }

be a

2

-bimodule. We dene the labelling

l

: V → {1, 2}

of

G

indu ed by

{X, Y }

. If

|X| = |Y | = 1

, then

l(x) = 1

and

l(y) = 2

, where

X

= {x}

and

Y

= {y}

. If

|X| = 1

, then

l(v) = 1

i

v

∈ N [x]

. Similarly if

|Y | = 1

,then

l(v) = 1

i

v

∈ N [y]

. Nowwe suppose

min(|X|, |Y |) > 1

. If

{X, Y }

isa split,then theset of verti es in

X

witha neighbour

Y

and theset of verti es in

Y

witha neighbour in

X

islabelled

1

,others verti es arelabelled

2

. If

{X, Y }

is a o-split, thena labelling of

G

indu ed by

{X, Y }

isalabellingof

G

indu edbythesplit

{X, Y }

. Finallyif

{X, Y }

isabi-join,

l

issu h that

{v ∈ X : l(v) = 1}

is a join with

{v ∈ Y : l(v) = 1}

and

{v ∈ X : l(v) = 2}

is a joinwith

{v ∈ Y : l(v) = 2}

. Notethatif

{X, Y }

isabi-join,thenthereistwopossibleslabelling

l

1

and

l

2

, with

l

1

= s(l

2

)

. If

{X, Y }

is a

2

-bimoduleof

G

and

l

a labelling indu edby

{X, Y }

,then every mono- olouredmodule hassize

1

(sin e

G

is primeand

|V | ≥ 3

).

Denition 19 (Good

2

-bimodule). A

2

-bimodule

{X, Y }

is good if the graph

G

with the labellingindu ed by

{X, Y }

isNLC-2

ρ

-free. Thefollowingproposition omes immediatelyfrom lemma3.

Proposition 20.

G

isNLC-2 if andonly if

G

has a good

2

-bimodule.

Lemma 21. If

G

has a good

2

-bimodule

{X, Y }

whi h is a split, then

G

has a good

2

-bimodule whi his a strong split.

Proof. There is a node

α

in the split de omposition tree and

∅ ( I ( {1, . . . , d(α)}

su h that

{X, Y } = {∪

i∈I

C

α

i

,

∪

i6∈I

C

α

i

}

. Let

l

: V → {1, 2}

bethelabellingof

G

indu edby

{X, Y }

. For all

i

∈ {1, . . . , d(α)}

,

(G[C

i

α

], l|

C

i

α

)

is NLC-2

ρ

-free (where

l|

W

isthe fun tion

l

restri tedat

W

). Let

l

′

be the

2

-labelling of

V

su h that for all

i

, and

v

∈ C

i

α

,

l(v) = 1

if and only if

v

has a neighbour outside of

C

i

α

. For all

i

, either

l|

C

i

α

= l

′

_|

C

i

α

, or

∀v ∈ C

i

α

,

l(v) = 2

. Then for all

i

,

(G[C

i

α

], l

′

|

C

i

α

)

isNLC-2

ρ

-free,andthus

(G, l

′

₎

isNLC-2

ρ

-free. Sin ethereisadominatingvertex inthe hara teristi graphof

α

, there is a

j

su h that the labelling indu ed bythe strong split

{C

α

j

, V

\ C

α

j

}

is

l

′

. Thus thestrong split

{C

j

Input: A graph

G

Result: Yes i

G

is NLC-2

S ←

thesetofstrong splits, o-splitsand bi-joinsof

G

; forea h

{X, Y } ∈ S

do

l

←

thelabellingof

G

indu edby

{X, Y }

;

if

(G[X], G[Y ], l)

is NLC-2

ρ

-free thenreturnYes ; returnNo ;

Algorithm 3: Re ognition of primeNLC-2graphs

Previouslemma on

G

saythatif

G

hasagood

2

-bimodule

{X, Y }

whi hisa o-split, then

G

hasagood

2

-bimodule whi his astrong o-split. The following lemma issimilar to Lemma 21. Lemma22. If

G

hasa good

2

-bimodule

{X, Y }

whi hisa bi-join,then

G

has agood

2

-bimodule whi his a strong bi-join.

Theorem 23. Algorithm 3 re ognises primeNLC-2 graphs, andits time omplexity is

O(n

2

_m)

.

Proof. Trivially if the algorithm return Yes, then

G

is NLC-2. On the other hand, by proposi-tion 20, and lemmas 21 and 22, if

G

is NLC-2, then it has a good strong

2

-bimodule and the algorithm returnsYes.

Theset

S

anbe omputed usingalgorithms for omputingsplitde ompositionon

G

and

G

, and bi-join de omposition on

G

. Note that it is not required to usea lineartime algorithm for splitde omposition [5 ℄: some simpler algorithms run in

O(n

2

_m)

[4, 10℄. [7, 8 ℄show that bi-join

de omposition anbe omputedinlineartime,usingaredu tiontomodularde omposition. But therealso,modularde ompositionalgorithmssimplerthan[15 ℄maybeused. Theset

S

has

O(n)

elements. Testingifa

2

-bimodule isgood takes

O(nm)

usingalgorithm 1. Sototal runningtime is

O(n

2

_m)

.

3.3 NLC-2 de omposition

Usinglemma 2,modular de omposition and algorithm 3,we get:

Theorem 24. NLC-2 graphs an be re ognised in

O(n

2

_m)

, and a NLC-2expression an be

gen-erated in the sametime.

4 Graph isomorphism on NLC-2 graphs

4.1 Graph Isomorphism on NLC-2

ρ

-free prime graphs

Thefollowingpropositionsaredire t onsequen esofproperties(linearanddegenerate)of

S

- uts. Proposition25. Considerasymmetri

S

∈ {1, 2}×{1, 2}

. Twographs

G

and

H

are isomorphi ifandonlyif there isa bije tion

π

between

P

S

(G)

and

P

S

(H)

su h that forall

P

∈ P

S

(G)

,

G[P ]

isisomorphi to

H[π(P )]

.

Proposition 26. Let a non-symmetri

S

∈ {1, 2} × {1, 2}

and let

G

and

H

be two graphs. Let

P

′

S

(G) = (P

1

, . . . , P

k

)

and

P

′

S

(H) = (P

1

′

, . . . , P

k

′

′

)

then

G

and

H

are isomorphi if and only if

k

= k

′

andfor all

i

∈ {1, . . . , k}

,

G[P

i

]

isisomorphi to

H[P

′

i

]

.

By theprevious 2 propositions,two NLC-2

ρ

-free 2-labelled graphs

G

and

H

areisomorphi ifand only ifthere is an isomorphism between their anoni al NLC-2

ρ

-free de omposition tree whi h respe ts the order of hildren of linearnodes. Thisisomorphism an be tested inlinear time,thus isomorphism of NLC-2

ρ

-free graphs anbe donein

O(nm)

time.

Input: Two primeNLC-2graphs

G

and

H

Result: Yes if

G

≃ H

,No otherwise

S ←

thesetofstrong splits, o-splitsand bi-joinsof

G

;

S

′

_←

thesetof strong splits, o-splitsand bi-joinsof

H

;

if there isno good

2

-bimodule in

S

thenfail with 

G

is notNLC-2;

{X, Y } ←

a good

2

-bimodulein

S

;

l

←

the labellingof

G

indu ed by

{X, Y }

; forea h

{X

′

_{, Y}

′

_{} ∈ S}

′

su h that

{X

′

_{, Y}

′

_}

is good do

l

′

←

thelabellingof

H

indu ed by

{X

′

_{, Y}

′

_}

;

if

|X| > 1

and

|Y | > 1

and

{X, Y }

isa bi-join then if

(G, l) ≃ (H, l

′

₎

or

(G, l) ≃ (H, s(l

′

₎₎

thenreturn Yes ; else if

(G, l) ≃ (H, l

′

₎

thenreturn Yes ;

returnNo ;

Algorithm 4: Isomorphismfor primeNLC-2graphs

4.2 Graph isomorphism on prime NLC-2 graphs

Theorem 27. Algorithm4 test isomorphismbetween two primeNLC-2 graphs intime

O(n

2

_m)

.

Proof. If the algorithm returns yes, then trivially

G

≃ H

. On the other hand suppose that

G

≃ H

andlet

π

: V (G) → V (H)

beabije tionsu hthat

{u, v} ∈ E(G)

i

(π(u), π(v)) ∈ E(H)

. Then

{X

′

_{, Y}

′

_}

with

X

′

_{= π(X)}

and

Y

′

_{= π(Y )}

is a good

2

-bimodule if

H

. If

min(|X|, |Y |) > 1

and

{X

′

_{, Y}

′

_}

isabi-join,thenbydenitionthereistwolabellingindu edby

{X, Y }

,and

(G, l) ≃

(H, l

′

)

or

(G, l) ≃ (H, s(l

′

₎₎

. Otherwise the labellingis unique and

(G, l) ≃ (H, l

′

₎

. Thesets

S

and

S

′

anbe omputedin

O(n

2

₎

timeusinglineartimealgorithmsfor omputing splitde omposition on

G

and

G

,and bi-joinde omposition on

G

. The sets

S

and

S

′

have

O(n)

elements. Testif a

2

-bimoduleis good take

O(nm)

using algorithm 1, and test iftwo

2

-labelled prime graphsareisomorphi take also

O(nm)

. Thus the total runningtimeis

O(n

2

_m)

.

4.3 Graph isomorphism on NLC-2 graphs

Itiseasy toshowthatgraphisomorphism onprimeNLC-2graphs withan additionallabels into

{1, . . . , q}

an be done in

O(n

2

_m)

time. For that, we add the additional label of

v

at the leaf orrespondingto

v

inthe NLC-2

ρ

-freede ompositiontree.

We show that we an do graph isomorphism on NLC-2 graphs in time

O(n

2

_m)

, using the modular de omposition and algorithm 4. Let

M(G)

and

M(H)

be themodular de omposition of

G

and

H

. For

M

∈ M(G)

, let

G

M

be

G[M ]

, and for

M

∈ M(H)

, let

H

M

be

H[M ]

. Let

G

∗

M

be the hara teristi graph of

G

M

(note that

|V (G

∗

M

)|

is the number of hildren of

M

in the modular de omposition tree). Let

M

(i,∗)

= {M ∈ M(G) ∪ M(H) : |M | = i}

, let

M

(∗,j)

= {M ∈ M(G) ∪ M(H) : |V (G

∗

M

)| = j}

and let

M

(i,j)

= M

(i,∗)

∩ M

(∗,j)

. Note that

P

n

j=1

(M

(∗,j)

× j)

is the number of verti es in

G

plus the number of edges inthe modular de omposition tree,and thus isat most

3n − 2

.

Theorem 28. Algorithm 5 testsisomorphism between two NLC-2 graphs in time

O(n

2

_m)

.

Proof. The orre tness omes from the fa t that at ea h step, for all

M, M

′

_{∈ M(G) ∪ M(H)}

su hthat

l(M )

and

l(M

′

₎

areset,

G

M

and

G

M

′

areisomorphi ifandonly if

l(M ) = l(M

′

₎

Input: Two NLC-2graphs

G

and

H

Result: Yes if

G

≃ H

,No otherwise

forevery

M

∈ M(G) ∪ M(H)

su h that

|M | = 1

do

l(M ) ← 1

; for

i

from

2

to

n

do

for

j

from

2

to

i

do

Compute the partition

P

of

M

(i,j)

su hthat

M

and

M

′

areinthesame lass of

P

ifand onlyif

(G

∗

M

, l) ≃ (G

∗

M

′

, l)

. ; forea h

P

∈ P

do

a

←

a newlabel(an integer not in

Img(l)

) ; For all

M

∈ P

,

l(M ) ← a

;

Algorithm 5: Isomorphism onNLC-2graphs

total time

f

(n, m)

of thisalgorithm is

O(n

2

_m)

sin e (big Oh isomitted):

f

(n, m) ≤

X

i

X

j

j

2

m

|M

(i,j)

|

2

≤ m

X

j

j

2

X

i

|M

(i,j)

|

2

!

≤ m

X

j

j

2

|M

(∗,j)

|

2

≤ m

X

j



j|M

(∗,j)

|

2



≤ n

2

m.

Referen es

[1℄ M.Chein,M.Habib,andM.C.Maurer. Partitivehypergraphs. Dis reteMath.,37(1):3550,1981. [2℄ B.Cour elle,J.Engelfriet,andG.Rozenberg. Handle-rewritinghypergraphgrammars. J. Comput.

Syst. S i.,46(2):218270,1993.

[3℄ W.H.CunnighamandJ.Edmonds.A ombinatorialde ompositiontheory.Canad.J.Math.,32:734 765,1980.

[4℄ William H. Cunningham. De omposition of dire ted graphs. SIAMJ. Algebrai Dis rete Methods, 3(2):214228,1982.

[5℄ E. Dahlhaus. Parallelalgorithms forhierar hi al lusteringand appli ationsto split de omposition andparitygraphre ognition. J.Algorithms,36(2):205240,2000.

[6℄ E.Dahlhaus,J.Gustedt,andR.M.M Connell. Partially omplementedrepresentationsofdigraphs. Dis reteMath.Theor.Comput. S i.,5(1):147168,2002.

[7℄ F. de Montgoler and M. Rao. The bi-join de omposition. In ICGT, volume 22 of ENDM, pages 173177,2005.

[8℄ F.deMontgolerandM.Rao. Bipartitivesfamiliesandthebi-joinde omposition. Te hni alreport, https://hal.ar hives-ouvertes.fr/hal-00132862,2005.

[9℄ J.-L. Fouquet, V. Giakoumakis, and J.-M. Vanherpe. Bipartite graphs totally de omposable by anoni alde omposition. Internat.J. Found.Comput.S i., 10(4):513533,1999.

[10℄ C.P.Gabor,K.J.Supowit,andW.-L.Hsu. Re ognizing ir legraphsinpolynomialtime. J.ACM, 36(3):435473,1989.

[11℄ T.Gallai. TransitivorientierbareGraphen. A taMath. A ad. S i.Hungar.,18:2566,1967.

[12℄ F. Gurskiand E. Wanke. Minimizing NLC-widthis NP-Complete. InWG,volume3787 ofLNCS, pages6980,2005.

[13℄ M.Habib,C.Paul,andL.Viennot. Partitionrenementte hniques: Aninterestingalgorithmi tool kit. Internat.J. Found. Comput.S i., 10(2):147170,1999.

[14℄ Ö.Johansson.NLC 2

-de ompositioninpolynomialtime.Internat.J.Found.Comput.S i.,11(3):373 395,2000.

[15℄ R. M. M Connell and J. P. Spinrad. Modular de omposition and transitiveorientation. Dis rete Math., 201(1-3):189241,1999.

A.1 Proof of lemma 13

Let

G

= (X, Y, E

j

, E

m

)

beaBTsu hthateveryBT-modulehassize

1

. Let

(x

1

, . . . , x

|X|

)

be

X

sorted by

(d

j

(x), d

m

(x))

inlexi ographi de reasing order. If

(A, B)

is a semi-joinof

G

,thenthere isa

k

∈ {0, . . . , |X|}

su h that

A

∩ X = {x

1

, . . . , x

k

}

.

Proof. For all

v

∈ A ∩ X

,

d

j

(v) ≥ |B ∩ Y |

, and for all

v

∈ B ∩ X

,

d

j

(v) ≤ |B ∩ Y |

. Moreover, if there is a

v

∈ B ∩ X

with

d

j

(v) = |B ∩ Y |

, then

d

m

(v) = 0

. Let

C

= {v ∈ X : d

j

(v) =

|B ∩ Y |

and

d

m

(v) = 0}

. Then

C

is a BT-module of

G

, and thus

|C| ≤ 1

. Every vertex in

A

∩ X \ C

arebeforeeveryvertexin

B

∩ X \ C

intheordering. Moreover,if

|C| > 0

,thenverti es in

A

∩ X \ C

arebeforethe vertexin

C

,and verti es in

B

∩ X \ C

are after thevertexin

C

in theordering.

A.2 Proof of lemma 14

Let

k

∈ {0, . . . , |X|}

and

k

′

_{∈ {0, . . . , |Y |}}

. Then

(A, (X ∪ Y ) \ A)

, where

A

=

{x

1

, . . . , x

k

, y

1

, . . . , y

k

′

}

,isasemi-joinof

G

ifandonlyif

P

k

i=1

d

j

(x

i

) −

P

k

′

i=1

d

j

(y

i

) =

k

× (|Y | − k

′

₎

and

P

k

i=1

d

m

(x

i

) −

P

k

′

i=1

d

m

(y

i

) = 0

.

Proof. TheIf partisbydenition. Now letus onsidertheOnly if part. Letus assumethat thedegree onditionholds. Wewilldenote

a

thenumberofjoinedgesbetween

A

∩ X

and

B

∩ Y

,

b

the numberofjoinedgesbetween

A

∩ X

and

A

∩ Y

,and

c

thenumber ofmixededges between

A

∩ X

and

A

∩ Y

. Note that

a

≤ k(|Y | − k

′

₎

,

a

+ b =

P

k

i=1

d

j

(x

i

)

and

b

≤

P

k

′

i=1

d

j

(y

i

)

, thus

a

≥ k(|Y | − k

′

)

. So we have

a

= k(|Y | − k

′

₎

,and

P

k

′

i=1

d

j

(y

i

) − b = 0

. In other words, there is onlyjoinedgesbetween

A

∩ X

and

B

∩ Y

,and thereisnojoinedges between

A

∩ Y

and

B

∩ X

. Now sin e there is only join edges between

A

∩ X

and

B

∩ Y

,

c

=

P

k

i=1

d

m

(x

i

) =

P

k

′

i=1

d

m

(y

i

)

, thus there isno mixededges between

A

∩ Y

and

B

∩ X

.

A.3 Algorithm to ompute

P

′

S

when

S

is non-symmetri

Proof. Corre tness: Algorithm6generatesallthesemi-joinsof

B

. Atanytime,

s

j

=

P

k

i=1

d

j

(x

i

)

,

s

m

=

P

k

_i=1

d

m

(x

i

)

,

s

′

j

=

P

k

′

i=1

d

j

(y

i

)

and

s

′

m

=

P

k

′

i=1

d

m

(y

i

)

. In

B

, every BT-modulehassize

1

, otherwisethereisamono- olouredmodulein

G

ofsizeatleast

2

. If

(A, B)

isasemi-join,thenby lemma13on

(C

1

,

C

2

, E

j

, E

m

)

and

(C

2

,

C

1

, E

j

, E

m

)

,thereisa

a

and

b

su hthat

A∩C

1

= {x

1

, . . . , x

a

}

and

A

∩ C

2

= {y

1

, . . . , y

b

}

. At anytime,

(A

′

_,

_(C

1

∪ C

2

) \ A

′

)

with

A

′

_{= {x}

1

, . . . , x

l

, y

1

, . . . , y

l

′

}

is the last semi-join found. At

k

= a

, the while line 12 will stop when

s

j

− s

′

j

= k × (|C

2

| − k

′

)

sin e for every

v

∈ A ∩ C

2

,

d

j

(v) ≤ k

,and

s

′

j

+ k × (|C

2

| − k

′

)

de rease with

k

′

. Moreover, when thewhileloop stops,

s

m

= s

′

m

sin e

s

′

m

in rease with

k

′

. Thus if

b

6= k

′

,then

{y

k

′

+1

, . . . y

b

}

isa BT-module and

b

= k

′

_{+ 1}

(sin e every BT-module has size

1

). In all ases the algorithm nds

(A, B)

,and addsthe partition in

P

′

.

Complexity: Aswe see inproof of theorem15, every instru tionlines [2-5℄ an be done in

linear time, and learly every instru tion lines [6-22℄ an be done inlinear time, thus the total runningtimeis

O(n + m)

.

Input: A

2

-labelled graph

G

,anda non-symmetri

S

⊆ {1, 2} × {1, 2}

Output:

P

′

S

V

i

← {v : v ∈ V

and

l(v) = i}

; 1

if

(1, 1) ∈ S

then

C

1

←

o- onne ted omponentsof

G[V

1

]

; 2

else

C

1

←

onne ted omponents of

G[V

1

]

; 3

if

(2, 2) ∈ S

then

C

2

←

o- onne ted omponentsof

G[V

2

]

; 4

else

C

2

←

onne ted omponents of

G[V

2

]

; 5

B = (C

1

,

C

2

, E

j

, E

m

) ←

thebipartite trigraphbetween theelements of

C

1

and

C

2

; 6

(x

1

, . . . , x

|C

1

|

) ← C

1

sorted by lexi ographi orderon

(−d

j

(v), −d

m

(v))

; 7

(y

1

, . . . , y

|C

2

|

) ← C

2

sortedbylexi ographi order on

(d

j

(v), d

m

(v))

; 8

P

′

_{← ()}

;

l

← 0

;

l

′

_{← 0}

;

k

′

_{← 0}

;

k

← 0

; 9

s

j

← 0

;

s

m

← 0

;

s

′

j

← 0

;

s

′

m

← 0

; 10 while

k

≤ |C

1

|

do 11 while

s

j

− s

′

j

< k

× (|C

2

| − k

′

)

or (

s

j

− s

′

j

= k × (|C

2

| − k

′

)

and

s

m

> s

′

m

) do 12

k

′

← k

′

_{+ 1}

;

s

′

j

← s

′

j

+ d

j

(y

k

′

)

;

s

′

m

← s

′

m

+ d

m

(y

k

′

)

; 13 if

s

j

− s

′

j

= k × (|C

2

| − k

′

)

and

s

m

= s

′

m

then 14

add

{x

l+1

, . . . , x

k

} ∪ {y

l

′

+1

. . . , y

k

′

}

at theend of

P

′

;

l

← k

;

l

′

_{← k}

′

; 15 if

s

j

− s

′

j

− d

j

(y

k+1

) = k × (|C

2

| − k

′

− 1)

and

s

m

= s

′

m

+ d

m

(y

k+1

)

then 16

k

′

← k

′

_{+ 1}

;

s

′

j

← s

′

j

+ d

j

(y

k

′

)

;

s

′

m

← s

′

m

+ d

m

(y

k

′

)

; 17

add

{y

k

′

}

at the end of

P

′

;

l

′

_{← k}

′

; 18

k

← k + 1

;

s

j

← s

j

+ d

j

(x

k

)

;

s

m

← s

m

+ d

m

(x

k

)

; 19 remove

∅

form

P

′

,ifany; 20 if

(2, 1) ∈ S

thenreverse

P

′

; 21 return

P

′

22 Algorithm 6: Computationof

P

′

S

when

S

is non-symmetri